Tensorization of the strong data processing inequality for quantum chi-square divergences

Yu Cao1 and Jianfeng Lu1,2

1Department of Mathematics, Duke University, Box 90320, Durham NC 27708, USA
2Department of Physics and Department of Chemistry, Duke University, Box 90320, Durham NC 27708, USA

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Abstract

It is well-known that any quantum channel $\mathcal{E}$ satisfies the data processing inequality (DPI), with respect to various divergences, e.g., quantum $\chi^2_{\kappa}$ divergences and quantum relative entropy. More specifically, the data processing inequality states that the divergence between two arbitrary quantum states $\rho$ and $\sigma$ does not increase under the action of any quantum channel $\mathcal{E}$. For a fixed channel $\mathcal{E}$ and a state $\sigma$, the divergence between output states $\mathcal{E}(\rho)$ and $\mathcal{E}(\sigma)$ might be strictly smaller than the divergence between input states $\rho$ and $\sigma$, which is characterized by the strong data processing inequality (SDPI). Among various input states $\rho$, the largest value of the rate of contraction is known as the SDPI constant. An important and widely studied property for classical channels is that SDPI constants tensorize. In this paper, we extend the tensorization property to the quantum regime: we establish the tensorization of SDPIs for the quantum $\chi^2_{\kappa_{1/2}}$ divergence for arbitrary quantum channels and also for a family of $\chi^2_{\kappa}$ divergences (with $\kappa \ge \kappa_{1/2}$) for arbitrary quantum-classical channels.

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